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It is given that the constant term in the expansion of
(2+(1)/(x))^(2)(1-3x)^(n) is 67 , where
n is a positive integer. Find the value of
n.

It is given that the constant term in the expansion of $\left(2+\frac{1}{x}\right)^{2}(1-3 x)^{n}$ is $67$ , where $n$ is a positive integer. Find the value of $n$ .

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Transformations of functions

Full solution

Q. It is given that the constant term in the expansion of

\left(2+\frac{1}{x}\right)^{2}(1-3 x)^{n}

is

67

, where

n

is a positive integer. Find the value of

n

.

Identify Terms Form: Identify the general form of the terms in the expansion of $(2 + 1/x)^2$ and $(1 - 3x)^n$ . For $(2 + 1/x)^2$ , expand using binomial theorem: $(2 + 1/x)^2 = 4 + 4/x + 1/x^2$ . For $(1 - 3x)^n$ , use binomial theorem: $(1 - 3x)^n = \Sigma (\text{from } k=0 \text{ to } n) [\binom{n}{k} \cdot 1^{(n-k)} \cdot (-3x)^k]$ .
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